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Annu. Rev. Fluid Mech. 1994.26 : 23-~63
Copyright © 1994 by Annual Reviews Inc. All rights reserved
LAGRANGI-AN PDF METHODS
FOR TURBULENT FLOWS
S. B. Pope
Sibley School of Mechanical and Aerospace Engineering,
Cornell University, Ithaca, NewYork 14853
KEYWORDS:
turbulence modeling, stochastic methods, Langevinequation
1.
INTRODUCTION
Lagrangian Probability Density Function (PDF) methods have arisen
the past 10 years as a union betweenPDFmethodsand stochastic Lagrangian
models, similar to those that have long been used to study turbulent
dispersion. The methods provide a computationally-tractable way of calculating the statistics,
of inhomogeneousturbulent flows of practical
importance, and are particularly attractive if chemical reactions are
involved. The information contained at this level of closure--equivalent
to a multi-time Lagrangian joint pdf--is considerably more than that
provided by momentclosures.
The computational implementation is conceptually simple and natural.
At a given time, the turbulent flow is represented by a large number
of particles, each having its ownset of properties--position, velocity,
composition etc. These properties evolve in time according to stochastic
model equations, so that the computational particles simulate fluid
particles. The particle-property time series contain information equivalent
to the multi-time Lagrangian joint pdf. But, at a fixed time, the ensemble
of particle properties contains no multi-point information: Each particle
can be considered to be sampled from a different realization of the flow.
(Hence two particles can have the same position, but different velocities
and compositions.)
It is generally acknowledged(e.g. Reynolds 1990) that manydifferent
approaches have important roles to play in tackling the problems posed
by turbulent flows. Each approach has its ownstrengths and weaknesses.
23
0066-4189/94/0115-0023505.00
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POPE
At one end of the spectrum of approaches, Direct Numerical Simulations
(DNS) offer unmatched accuracy, but their computational cost is high,
and their range of applicability is extremely limited (Reynolds 1990).
the other .end of the spectrum, simple turbulence models such as k-~
(Launder & Spalding 1972) offer essentially unrestricted applicability,
moderate cost, but poor or uncertain accuracy (except in some simple
flows). Compared to suclh turbulence models, Lagrangian PDFmethods
have the same wide applicability; their cost is greater (but less than DNS);
and, for the reasons presented below, their potential for accuracy is greatly
increased.
In going beyond simple turbulence models (i.e. momentclosures), the
challenge is to incorporate a fuller description of the turbulence, while
retaining computational tractability. If appropriately chosen, the fuller
description allows unsatisfactory modeling assumptions to be avoided,
and at the same time provides more information to model the unavoidable.
It can be argued that the dominant process in turbulent flows is convection (by the instantaneous fluid velocity). At high Reynolds number,
molecular diffusion makes a negligible contribution to spatial transport,
and so convection dominates the transport of momentum, chemical
species, and enthalpy. In momentclosures, at some level, convection is
modeled by a gradient diffusion assumption, which can lead to qualitatively incorrect behavior (see, for example, Deardorff 1978). In Lagrangian
PDFmethods, on the other hand, convection is treated simply and naturally in the Lagrangian frame with no modeling assumptions. Similarly, in
reacting flows, finite-rate nonlinear reaction rates can present insurmountable difficulties to momentclosures; whereas arbitrarily complex
reactions can be handled naturally by Lagrangian PDFmethods, without
modeling assumptions (Pope 1985, 1990).
In any statistical
approach to turbulence, modeling assumptions are
required at somelevel. It is notoriously difficult to construct general and
accurate models for turbulence, whereas there are manyother stochastic
physical phenomenafor which simple statistical models are successful (see
e.g. van Kampen1983). There are two features of turbulent flows that go
a long way to explaining the inherent difficulties. First, turbulence has a
long memory:In free shear flows, it is readily deduced (from experimental
data) that the characteristic time scale of the energy-containing turbulent
motion is, typically, four times the characteristic mean-flowtime scale.
Second,through the fluctuating pressure field, the velocity field experiences
long-range interactions. Amongother effects, this can lead to large-scale
organized motions, and to the boundary geometry influencing the turbulence structure in the interior of the flow.
Lagrangian PDFmethods can take full account of the long memoryof
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LAGRANGIAN
PDF
METHODS
25
turbulence. For the fluid properties considered, the multi-time Lagrangian
joint pdf completelydescribes the past history of all fluid particles that (on
different realizations) pass through a given point at a given time. As
discussed in Section 5, the long-memoryincorporated in current stochastic
Lagrangian modelsleads to fluid-particle motions that are consistent with
large-scale turbulent structures.
In the next Section the relevant Eulerian and Lagrangian pdfs are
introduced, and the different PDFmethods are categorized. Section 3 is
devoted to the Langevinequation. This equation provides a simple stochastic modelfor the velocity of a fluid particle: It is also a building block for
other stochastic models. The particle representation of a turbulent flow,
which is fundamental to the Lagrangian PDFapproach, is described in
Section 4. Then more recent and sophisticated stochastic models are
reviewed in Section 5.
2.
EULERIAN
AND
LAGRANGIAN
PDF
METHODS
The purpose of this section is to introduce the various pdfs considered,
to categorize PDFmethods, and to provide references to the relevant
literature.
2.1
Eulerian
pdfs
Westart by considering a single composition variable (e.g. a species mass
fraction) which, at position x and time t, is denoted by ~b(x, t). At fixed
(x, t), ~b is a randomvariable, corresponding to which we introduce the
independent sample-space variable O. Then the cumulative distribution
function (cdf) of q~ is defined
F~(0, x, t) = Prob{q~(x, t) < 0},
(1)
and the probability density function (pdf) of q5
(2)
f~(0; x, t) = ~ F~(0, x, t).
The fundamentalsignificance of the pdf is that it measuresthe probability
of the randomvariable being in any specified interval. For example, for
0b > 0a, from Equations (1) and (2) we obtain
Prob
{0a
N
{#(X, t) < 0U} = f*(0; X, t)
(3)
a
The pdfjust defined, f,(x, t), is the one-point, one-time Eulerian pdf
~b(x, t). It completely describes the randomvariable q~ at each x and
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separately; but it contains no joint information about ~b at two or more
space-time points. If ~b(x, t) is statistically homogeneous,
then fo is independent of x.
Moregenerally, we maywant to consider a set oftr composition variables
~b(x, t) where~ = {qbl, q~2..... q~}.Then,with~k = {~b1, ~/~2, ¯ ¯ ¯, ~b~}being
corresponding sample-space variables, the joint pdf of ~b is denoted by
f~0P; x, t). Further, with U(x, t) being the Eulerian velocity of the fluid,
introduce sample-space velocity variables V = { V~, Vz, V3}and denote the
(one-point, one-time Eulerian) joint pdf of velocity by fu(V; x, t). Finally,
the velocity-compositionjoint pdf is denotedby f(V, ~p; x, t).
By definition, in a PDFmethod, a pdf (or joint pdf) in a turbulent flow
is determinedas the solution of a modeledevolution equation.
2.2
Assumed
PDF Methods
In spite of their name, assumed PDF methods are not PDF methods
(according to the above definition). Instead of being determined from
modeled evolution equation, the pdf is assumedto have a particular shape
that is parametrized (usually) by its first and second moments.The method
has found application in combustion (e.g. Bilger 1980). For the pdf of
single composition, the suggested shapes include: a beta-function distribution (Rhodes 1975); a clipped Gaussian (Lockwood& Naguib 1975);
and a maximum
entropy distribution (Pope 1980). The extension to several
composition variables, which is considerably more difficult, has been considered by, among others, Correa et al (1984), Bockhorn (1990),
Girimaji (1991).
Although assumed PDF methods are favored in some applications,
compared to PDFmethods they have two disadvantages. First, no account
is taken of the influence of the dynamics(e.g. reaction) on the shape of the
pdf. Second--and maybe surprising at first sight--assumed PDFmethods
are computationally more expensive (if not intractable) for the general
case of manycompositions.
2.3 Eulerian
PDF Methods
One of the first cases studied using PDFmethods, and one that continues
to receive considerable attention, is reaction in constant-density homogeneous turbulence. In the simplest situation, the composition is characterized by a single passive scalar 4,(x, t), that evolves
D~b_ FV2q
~ + S(x, t).
Dt
(4)
Here D/Dt = O/Ot+U"V is the substantial derivative, F is the (constant)
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LAGRANGIAN PDF METHODS
molecular diffusivity,
composition:
27
and the reaction rate is a knownfunction of the
s(x,t) = ~[~(x,t)].
(5)
For the statistically homogeneouscase considered, the composition pdf
f,($; t) is independent of x. Without further assumption, the evolution
equation for f, can be deduced from Equation (4). This can be done using
any one of several different techniques that have been developed over the
years. These techniques are reviewed by Pope (1985), Dopazo(1993),
Kuznetsov & Sabel’nikov (1990), and are not described here. For the
present case the result forf,(~k; t)
f,~ - -- ~02[/e~Z(~, t)]-- ~[f~(4’)],
(6)
where
X(~’, t) = F<Vq5
.VqSl~b(x,t) = ~b>
(7)
is the conditional scalar dissipation.
An important observation is that the reaction term in the pdf equation
(6) is in closed form, whereas the corresponding terms in momentclosures
[e.g. <~(~b)> and <~b3(40>]are not--hence the attraction of PDFmethods
for reactive flows.
The term involving Z in Equation (6) represents molecular mixing and
requires modeling. In general--as exemplified by Equation (7)--in
Eulerian PDFmethods, the quantities that have to be modeled are onepoint one-time conditional expectations.
Molecular mixing models (which model the term in Z in Equation 6)
have a long history which is briefly reviewed in Section 5.5 and more
thoroughly elsewhere (Pope 1982, 1985; Borghi 1988; Dopazo 1993); and
there are several relevant recent.works (Chert et al 1989, Sinai & Yakhot
1989, Valifio & Dopazo 1991, Pope 1991a, Gao 1991, Gao & O’Brien
1991, Fox 1992, Pope & Ching 1993).
The composition PDFmethod can be extended to several compositions,
and to inhomogeneousflows. The latter necessitates modeling turbulent
convection--generally as gradient diffusion. Recent applications to multidimensional flows are described by Chen et al (1990), Roekaerts (1991),
and Hsuet al (1993).
For inhomogeneous flows, the method based on the velocity-composition joint pdf f(V, q,; x, t) has the advantage of avoiding gradientdiffusion modeling. For constant-density flow, the Navier-Stokes equations can be written
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28
POPE
DU
Dt - vV2U- V(p) Vp’,
(8)
wherev is the kinematic viscosity, and the pressure (divided by the density)
p is subjected to the Reynolds decomposition. From this equation, and
from Equation (4) written for each composition qS~(x, t), the evolution
equation forf(V, ~k; x, t) can be deducedto
Of Vi~x~+~[f~(~)
] oQ~) of
~ +
~[f(~x~_VV
U,.lv,~p)]
Ox~~V~
-~[f(vV~v,
0)],
(9)
where (V~]V,O) is written
for the conditional
expectation
(Vz~]U(x, t) = V, ~(x, t) - ~), and the summationconvention applies
a as well as to i.
The terms on the left-hand side of Equation (9) are in closed form and
represent convection, reaction, and acceleration due to the meanpressure
gradient. Those on the right-hand side contain one-point one-time conditional expectations. Modelsfor the effects of the viscous stresses and the
fluctuating pressure gradient are discussed below.
2.4
Lagrangian pdfs
Fundamentalto the Lagrangiandescription is the notion of a fluid particle.
Let t0 be a reference time, and let x0 = {Xo~,Xoz, Xo~} be Lagrangiancoordinates. Thenx+(t, x0) denotes the position at time t of the fluid particle
that is at x0 at time t0 [i.e. x+(t0, x0) = x0].
For each Eulerian variable [e.g. U(x, t)] t~e corresponding Lagrangian
variable (denoted by the superscript +) is defined by (for example)
U+(t,x0)=U[x+(t,x0),
(10)
By definition, a fluid particle moveswith its ownvelocity. So, given the
Eulerian velocity, x+ is determined as the solution of
ex+(t, x0)
- ~+(~, *o),
Ot
(~
with the initial condition
(12)
x+(t0, x0) = x0.
To simplify the subsequent development we impose two restrictions.
First, we consider flow domains that are (possibly time-dependent)
material volumes. Thus fluid particles do not cross the boundary of the
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LAGRANGIAN PDF METHODS
29
flow domain. Second, we consider constant-density flow so that the determinant of the Jacobian ~x~-/dXoj is unity. Both of these restrictions are
readily removed(see Pope 1985).
The primary Lagrangian pdf considered is
(13)
fL(V, X; tlV0, x0),
whichis the joint pdf of the event
(14)
{U+(t, x0) = V, x+(/, Xo)
subject to the condition U+(t0,x0) = V0. Thus fL is the joint pdf of the
fluid particle properties at time t, conditional upon their properties at
time to. Note that, in the Lagrangian pdffL, x denotes the sample space
corresponding to x+(t, x0), while in the Eulerian pdf it is a parameter.
Also, other fluid particle properties [e.g. 4~+(t, x0)] can be included in the
definition offL.
The Lagrangian pdffL is defined in terms of the reference initial time
to and a future time t > to. More generally, we may consider Mtimes
to < t~ < t2... < tM, and define the M-timeLagrangian pdf
fLM(VM,XM;tin: VM- 1, XM-- 1; t~t
1;’’"
;
Vl,
Xl;
tl [Vo, x0)
(15)
as the joint pdf of the events
{U+(tk, x0) = Vk, x+(tk, x0) = x~; k = 1,2,...,
(16)
subject to the sameinitial condition as before, U÷(t0, x0) = V0.
2.5
Lagrangian
PDF Methods
In Eulerian PDFmethods, the quantities to be modeled are one-point,
one-time conditional expectations (see Equation 9). In Lagrangian PDF
methods, the modeling approach is entirely different. Stochastic models
are constructed to simulate the evolution of fluid particle properties. For
example, Figure 1 shows the time series of one component of velocity
(in stationary, homogeneous,isotropic turbulence) according to a simple
stochastic model--the Langevinequation (which is the subject of the next
Section).
It is useful to distinguish betweenthe fluid-particle properties (U÷ and
÷)
x and the values obtained from the stochastic models. Thus U*(t) and
x*(t) denote the modeledparticle properties, with x* evolving
dx*(t)
dt - U*(t).
(17)
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~ 0
-1
0
5
10
Figure 1 Sampleof an Ornstein-Uhlenbeck process obtained as the solution of the Langevin
equation (Equation 21).
Then fL*(V,X;tIV0, x0) is the joint pdf to U*(t) and x*(t) subject
initial condition
U*(t0) = Vo, x*(to) =
(18)
If the stochastic modelis accurate, then fL* is an accurate approximation
tofL.
Stochastic Lagrangian models--such as the Langevin equation--have
long been used in studies of turbulent dispersion, where the quantities of
interest are (or can be obtained from) Lagrangian pdfs. For example, if
pulse of a contaminant is released at time to and location x0, then (if
molecular diffusion can be neglected) the expected concentration of the
contaminantat a later time t is proportional to the pdf of x+(t, x0), which
is modeled by x*(t). In fact, as early as 1921, G. I. Taylor proposed
stochastic modelfor x*(t) precisely for this application (Taylor 1921).
A direct numerical implementation of a stochastic model of turbulent
dispersion is to release a large numberN of particles at the source [i.e.
x*(to) = Xo] with initial velocities U*(to) distributed according to
Eulerian pdff(V; Xo, to). Then the stochastic model equations are inte-
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LAGRANGIAN
PDF
METHODS
31
grated forward in time to obtain U*(t) and x*(t). The expected particle
numberdensity (at any x, t) is then proportional to the meancontaminant
concentration.
In Lagrangian PDFmethods, the stochastic models are used to determine both Lagrangian and Eulerian pdfs. This is achieved through the
fundamental relation:
f(V; x, t)
fff(vo;
x0
, t0)fL(V, X;tlV0, x0) dV0dx0,(19)
where integration is over all velocities and over the entire flow domainat
to. (A derivation of this equation is given by Pope1985.) Thusthe Lagrangian
pdffL is the transition density for the turbulent flow: It determines the
transition of the Eulerian pdf from time to to time t.
Since fL determines f, it also determines simple Eulerian means, such
as the mean velocity (U(x, t)) and the Reynolds stresses (uiuj) (where
u = U-(U)). Such means can therefore be used as coefficients in the
stochastic Lagrangian models.
It is almost inevitable that computationallyviable stochastic modelsare
Markovprocesses. That is, with t,_ 1 < t, < t,+ t, the joint pdf of U*(t,+ l)
and x*(t~+ l) is completely determined by U*(t,), x*(t,) and the Eulerian
pdff(V; x, t,), independentof the particle properties at earlier times t,_
It then follows that (the model equivalent of) the M-timeLagrangian pdf
(Equation 15) is given by the product of the Mtransition densities
fL*u(V~,xu; tu: V~t_1, XMl; tu_ ~;... ; V~,Xl; t~ ]Vo,Xo)
M
= I-I fL*(Vk, Xk;tklV,_ l, Xk_1). (20)
k=l
Thusfor Markovmodels it is sufficient to consider the transition density
J~* since this contains the same information as the M-timeLagrangian pdf.
Lagrangian PDFmethods are implemented numerically as Monte Carlo/
particle methods. In contrast to implementations for dispersion studies,
the large numberof particles are at all times uniformly distributed in the
flow domain. This particle representation is described in more detail in
Section 4. Calculations based on this approach are described by Haworth
& Pope (1987), Anand et al (1989, 1993), Haworth & E1 Tahry (1991),
Taing et al (1993), and Norris (1993), for example.
3.
LANGEVIN
EQUATION
The Langevin equation is the prototypical stochastic model. The basic
mathematicaland physical concepts are introduced here in a simple setting.
More general and advanced models are described in Section 5.
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Webegin by considering stationary homogeneousisotropic turbulence,
with zero meanvelocity, turbulence intensity u’, and Lagrangian integral
time scale T. The subject of the Langevinequation, U*(t), is a modelfor
one componentof the fluid-particle velocity U÷(t).
Written as a stochastic differential equation (sde), the Langevinequation
is
dU*(t) = -- U*(t) dt/T+ (2u’2/T)1/2 dl, V(t),
(21)
where W(t) is a Wiener process. The reader unfamiliar with sdes, can
appreciate the meaningof the Langevinequation through the finite-difference approximation
U*(t + At) = U*(t)- U*(t)At/T+ (2u’2At/T)~/2~,
(22)
where ~ is a standardized Gaussian random variable ((3) = 0, (~2)
which is independent of the corresponding random variable on all
other time steps. Thus the increment in the Wiener process dW(t) can be
thought of as a Gaussian random variable with mean zero, and variance
dr.
The basic mathematical properties of the Langevin equation are now
described, and then their relationship to the physics of turbulence is discussed.
The Langevin equation (Equation 21 or 22) describes a Markovprocess
U*(I) that is continuous in time (see Gardiner 1990, for a more precise
statement). Hence, in the terminology of stochastic processes, U*(t) is a
diffusion process. Althoughit is continuous, it is readily seen that U*(t) is
not differentiable: Equation (22) shows that [U*(t+At)-U*(t)]/At varies
as At- 1/2, and hence does not converge as At tends to zero.
For simplicity we consider the initial condition at time to that U*(to) is
a Gaussian random variable with zero mean and variance u’2. Then, for
t > to, U*(t) is the stationary random process knownas the OrnsteinUhlenbeck (OU) process, a sample of which is shown on Figure 1. The
OUprocess is a stationary,
Gaussian, Markov process, and hence is
completely characterized
by its mean ((U*(t))=0),
its variance
2)
((U*(t) = u’2), and its autocorrelation function, which is
p*(s) =( U*(t +s)U*(t) ’ ~ =e-I*I/T,
(23)
(see e.g. Gardiner 1990). Notice that these results confirm the consistency of the specification of the coefficients in the Langevin equation:
The rms fluid-particle velocity is u’, and the Lagrangian integral time
scale is
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LAGRANGIAN PDF METHODS
T = ff p*(s) ds.
33
(24)
To what extent does U*(t) model the fluid particle velocity U÷(t)? The
first and obvious limitation is that U÷(t) is differentiable, whereasU*(t)
is not. Hencethe modelis qualitatively incorrect if U*(t) is examinedon
an infinitesimal time scale.
But consider high Reynoldsnumberturbulence in which there is a large
separation between the integral time scale T and the Kolmogorovtime
scale %; and let us examineU+(t)on inertial-range time scales s, T >>s >>
This is best done through the Lagrangian structure function (see e.g.
Monin & Yaglom 1975)
DL(s) = ([U+(t+s) - U+(t)]2).
(25)
The Kolmogorovhypotheses (both original 1941 and refined 1962) predict
(in the inertial range)
DL(s) = Co(e)s,
(26)
whereCo is a universal constant, and (e) is the meandissipation rate. And
the Langevinequation yields [for the structure function based on U*(t)]:
De*(s) = 2u’2s/T, for siT << 1,
(27)
as is evident from Equation (22). Thus the Langevinequation is consistent
with the Kolmogorovhypotheses in yielding a linear dependenceof De on
s in the inertial range. (Equation 27 corresponds to an ~o-z frequency
spectrum (at high frequency), which in turn corresponds to white-noise
acceleration.)
By comparingthe coefficients in Equations (26) and (27) we obtain
relation
T-l
=
Co(a)/(2u ’2) = ~Co(e>/k,
(28)
where k is the turbulent kinetic energy; and the Langevin equation
(Equation 21) can be rewritten in the alternative form:
dU*(t) = - ~ Co U*(t) dt (C0(e>)1/2 dW(t).
(29)
To date, Lagrangian statistics
in high-Reynolds number flows have
proven inaccessible both to experiment and to direct numerical simulation.
Consequently, a direct test of Equation (26) has not been possible. However at low or moderate Reynolds number, both techniques have been
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used to measure the Lagrangian autocorrdation function p(~’). Figure
shows the results comparedto the exponential (Equation 23) arising from
the Langevin equation. At very small times s/T, the behavior is qualitatively different because U*(t) is not differentiable--correspondingly,
p*(s) has negative slope at the origin. But for larger times, the exponential
form provides a very reasonable approximation to the observed autocorrelations.
Measurementson turbulent dispersion provide an indirect test of the
Langevinequation. For particles originating from the origin at time t = 0,
their subsequent position is
(30)
x*(t) = U*(t’)
1.0
0.9
0.8
¯ ~ Expt. Sato & Y~mamoto (1987)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
s/T
Figure 2 Lagrangian velocity autocorrelation function in isotropic turbulence: open symbols and lines from DNSof Yeung & Pope 0989); full symbols from experiments of Sato
Yamamoto(1987); dashed line exponential, Equation 23. (From Yeung & Pope 1989.)
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LAGRANGIAN PDF METHODS
35
This (according to the Langevinequation) is a Gaussian process, with zero
mean, and variance
(x*(t)2~ 2u’2T[t- T(1 -- e-’/r)],
(31)
which exhibits the correct short-time limit [(x*(t) 2) ~ (u’t) 2] and longtime limit [(x*(t) 2) ~ 2u’2Tt] given by Taylor’s (1921) theory.
The Langevin equation has been applied to dispersion behind a line
source in grid turbulence by Anand & Pope 0985) with modifications to
account for the decay of the turbulence and the first-order effects of
molecular diffusion. The result shownon Figure 3 is in excellent agreement
with the data.
Several refinements and extensions to the Langevin equation are
described in Section 5, where further comparisons with DNSdata are
made.
Wenow describe the most rudimentary extension of the Langevin equation to inhomogeneousflows. The equation is a model for the evolution
of all three componentsof velocity U*(t) of a fluid particle with position
X*(t).
’
I
~
I
’
I
’
I
o.~,/
10°
z’//o
~
~
-I
10
/
"~/
05
.2
10
~~~ ~"lJ
"~
310
,
~
la,""
I
"2
10
¯ -
=
X
+
--52
= 60
Stapountzis et.al.
[]-’~o/M ~ 19.3
,
I
110
,
I
°10
,
I
101
~w /~o
Figure 3 Turbulent dispersion behind a line source in grid turbulence. The rms dispersion
x’ -- (x .2) ~/2 is normalized by the integral scale at the source lo; the distance downstream
of the source xw is normalized by the distance from the grid to the source Xo. Experimental
data of Warhaft(1984) and of Stapountzis et al (1986). Solid line is the calculation of Anand
& Pope (1985) based on the Langevin equation.
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There are three modifications to Equation (29)--all to the drift term.
First, the velocity increment due to the mean pressure gradient -dtV(p)
is added. Second, the fluid particle velocity relaxes to the local Eulerian
mean(U(x*[t], t)) (rather than to zero). And,third, the coefficient
drift term is altered. Theresult is
+(Co(~))’/2dw(t).(32)
In this (and subsequent) equations, it is understood that meanquantities
(i.e. V(.p), (~), and (U)) ar e evaluated at theflui d-particle posi tion
x*(t). The vector-valued Wiener process W(t) is simply composedof three
independent components W~(t), Wz(t), and W3(t ). The increment dWhas
zero mean and covariance
( dWidW/) = dt 6ii.
(33)
[As discussed at greater length in Section 5.1, the coefficient (½+ ~C0)
Equation (32) (compared to ]C0 in Equation 29), correctly causes
turbulent kinetic energy to be dissipated at the rate (~). The omission
the ½ in Equation (29) is because that equation pertains to the hypothetical
case of stationary (i.e. non-decaying)isotropic turbulence.]
A stochastic modelfor fluid-particle properties implies a modeledevolution equation for the corresponding Lagrangian joint pdf. In the present
context, f~*(V,x, tlVo, x0)is the joint pdf of U*(t) and x*(t), with
initial conditions U*(t0) =: V0, x*(t0) = x0- Thenwith U*(t) evolving
extended Langevin equation (Equation 32), and with x*(t) evolving
Equation (17), fL* evolves according to the Fokker-Planck equation
O ¯ -- Vi Of L* O@)
~fC = ~x~ + ~xx~
~
+
O4)
(see Gardiner 1990, Risken 1989), with the initial condition
fY(V,x, t01v0,x0)= a(v-v0)a(x-
05)
The Eulerian joint pdf of velocity f(V; x, t) is related to its Lagrangian
counterpart by Equation (19). Hence the above evolution equation forf~
implies a corresponding evolution equation for the modeled Eulerian pdf
f*(V; x, 0, Indeed, sincc the differential operators in the Fokker-Planck
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equation are independent of to, x0, and V0, it follows immediately from
Equation (19) that the Eulerian pdfJ’*(V; x, t) also evolves according
Equation (34).
The Eulerian pdf equation (i.e. Equation 34 written for f*), together
with a modeled equation for (e), form a complete set of turbulencemodelequations. Theyare completein the sense that all the coefficients in
Equation (34) are knownin terms off* and (e.). The mean velocity
and the kinetic energy k are determined as first and second momentsof
f*, while the meanpressure field (p) is determined as the solution of
Poisson equation. The source in the Poisson equation involves (U) and
(uiuj) which are knownin terms off*.
In Section 5.3 coupled stochastic models for U*(t) and 09*(0 = e*(t)/k
are described, which lead to an evolution equation for the joint pdf of U
and ~o. This single equation provides a complete model: All of the
coefficients are knownin terms of the pdf itself.
The above development illustrates
the different use of the Langevin
equation in turbulent dispersion and in PDFmethods. In the former, the
turbulent flow field is assumed known, and so the coefficients in the
Langevin equation are specified; and the equation is used (at most)
deduce the Lagrangian pdf. In PDF methods, the Langevin equation
is used to determine the Eulerian pdf, from which the coeff~cients are
deduced.
4.
PARTICLE
REPRESENTATION
Central to Lagrangian PDFmethods is the idea that a turbulent flow can
be represented by an ensemble of N fluid particles, with positions and
velocities x~n)(t), U~n)(t), n = 1, 2 .... , N. Thepurposeof this section
describe this particle representation, and to make precise the connection
betweenparticle properties and statistics of the flow.
4.1
Basic Representation
Webegin by considering a single componentof velocity U at a particular
point and time. Thus U is a random variable, with pdff(V), which
consider to be known.
For a given ensemblesize N(N >_ 1), the particle velocities {U~")} are
specified to be independent random samples, each with pdff(V). It
conceptually useful (and legitimate) to think of ~") as t he value of Uon
the n-th (independent) realization of the flow. Note that the particle velocities are independent and identically distributed, and hence the numbering of the particles is irrelevant.
A fundamental question, to which we provide three answers, is: In what
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sense does the ensemble (Ut")) "represent" the.underlying distribution
f(V)?
The first answer is in terms of the discrete pdffN(V), defined
1
fr~(V) =- ~ ~ 6(U~")- V).
(36)
It is readily shown(see, e.g. Pope 1985) that the expected discrete pdf
equalsf(V) for any N > l:
(37)
<fr~(V)) = f(V).
In the analysis of PDFmethods, this relation allows properties of the pdf
[i.e. f(V)] to be deducedfrom the properties of a single particle (~), say).
The second answer involves ensemble averages. In numerical implementations of PDFmethods it is necessary to estimate means such as (U) and
(U2) from the ensemble{ U(")}. Let Q(U)be somefunction of the velocity
U, then we have
(38)
<Q(U)) = f~_~ f(V)Q(V)
For example, the choices of V and Vz for Q(V) lead to (U) and (U:).
The mean (Q) can be estimated from the ensemble simply as the ensemble
average
<Q(U)>u --
N Q(O~"))
= fu(V)Q(V)
(39)
Then (since {U~")} are independent and identically distributed) a basic
result from statistics is that <Q>uis an unbiased estimator of <Q>:
<<Q>N>= <Q>-
(40)
Further, if the variance of Q(U)is finite, it follows from the central limit
theoremthat for large N the rms statistical error in <Q>utends to zero as
N ~/2
Hence the second sense in which the ensemble { Ut"~} "represents" the
pdff(V) is that, for all functions Q [for which Q(U) has finite meanand
variance], the ensemble average <Q>uconverges in mean square to <Q>.
This is written
lim <Q>u= <Q>.
(41)
[An additional convergence result is provided by the Glivenko-Cantelli
theorem (e.g. Billingsley 1986): As N tends to infinity, the difference
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betweenthe cdf F( IO and the empirical cdfF,( V)--i.e. the definite integral
offN(V)--converges to zero with probability one.]
For almost all purposes, the two answers provided above are sufficient:
Equation (37) is used in the analysis of PDFmethods, while Equation (41)
is used in numerical implementations. However,neither of these relations
(nor the Glivenko-Cantelli theorem) provides an estimate of the pdff(V)
in terms of { Uen)) that converges in meansquare as N tends to infinity.
The third answer, then, is that the techniques of density estimation can be
used for this purpose (see e.g. Tapia & Thompson1978, Silverman 1986).
These are not reviewed here, since they have not played an important role
in PDFmethods. This is because in the implementation of PDFmethods,
an explicit representation of the pdf is not required.
The above considerations apply to any random variable U. Consider
nowU¢n)(t) to be a modelfor the velocity of a fluid particle, obtained as
the solution to a stochastic model equation--the Langevin equation, for
example. At the initial time to, the values of (U~")(to)) are sampled from
the specified initial pdff(V; to).
The representations described above are readily extended. The one-time
discrete pdf [representingf(V; t)]
1 u
fu( V; t) = ~ .~=, 6[U~")(t)--
(42)
while the discrete Lagrangian pdf is
1 u
f~N(V;tl V0) = ~ ,~1 {3[U~")(t)- V]I U~")(to) = V0}.
Multi-time Lagrangian statistics
for example,
can be estimated as ensemble averages:
1 u
(Q(U(tl), U(t2)))u = ~ ,~l Q[U~")(tO’U~")(t2)]"
4.2
Inhomogeneous
(43)
(44)
Flows
The extension of this particle representation to inhomogeneous flows
requires some new ingredients, and it leads to some subtle consistency
conditions.
Throughout,for simplicity, we are restricting our attention to constantdensity flows in a material volume. Hence the volumeV of the flow domain
D, and the mass of fluid within it, do not changewith time.
At a given time t, an ensembleof N particles is constructed as follows
to represent the joint pdf of velocityf(V; x, t). Theparticle positions x~")(t)
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are mutually independent, random, uniformly-distributed in D. [Hence the
pdf of each x~n)(t) is l/V.] Thenthe particle velocity U~)(t) is random,
pdff[V; x~n)(t), t]. In terms of these properties, the discrete pdf is defined
by
fN(V;x, t) --= ~ 6[x~(t) -- xlf[U~")(t)--
(45)
i=1
The specification of x~)(t) (and also the constant in Equation 45)
determined by a consistency condition. Werequire the expectation offN
to equal f; where f satisfies the normalization condition that its integral
over all V is unity. Hence from Equation (45) we obtain
1 = f(f~v)
dV = V(f[x(")(t)-
x]),
for any n (since {x~")} are independent and identically distributed). This
condition is satisfied if, and only if, x(")(t) is uniformlydistributed.
If this consistencyconditionis satisfied at an initial time to, will it remain
satisfied as the particle properties evolve in time? The answer (established
by Pope 1985, 1987) is yes, provided the mean continuity equation is
satisfied. This in turn requires that the meanpressure gradient [affecting
the evolution of U(")(t), Equation 32] satisfies the appropriate Poisson
equation.
Both of these results are reflected in the Eulerian pdf equation [e.g.
Equation 34 written for f(V; x, t)]. Whenthis equation is integrated over
all V, all the terms on the right-hand side vanish, expect the first whichis
-V" (U). If this is nonzero--in violation of the continuity equation-then the normalization condition on f is also violated. An evolution
equation for V" (U) is obtained from Equation (34) by multiplying by
integrating over all V, and then differentiating with respect to xj. Equating
the time rate of change of V" (U) to zero, yields a Poisson equation for
@).
With Q(V) being a function of the velocity, we now consider the estimation of the mean (Q[U(x, t)]) from the ensemble of Particles. This
an important issue because Eulerian means such as (U) and (uiuj) must
be estimated from the particle properties in order to determine the
coefficients in the modeledparticle evolution equations, e.g. Equation (32).
Since (with probability one) there are no particles located at x, it
unavoidable that an estimate of (Q[U(x, t)]) must involve particles in
vicinity of x. Wedescribe nowthe kernel estimator (see e.g. Eubank1988,
H/irdle 1990), which is useful both conceptually and in practice (although
a literal implementation is not efficient). It is assumedthat Q[U(x, t)]
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has finite mean and variance, and that the mean is twice continuously
differentiable with respect to x.
For simplicity we consider points x that are remote from the boundary
of the domain; and for definiteness we take the kernel to be a Gaussian of
specified width h. In D dimensionsthis is
K(r, h) = (x/~h)-° exp (- ½r2/h2),
(47)
(wherer = Irl). Then a kernel estimator of (Q[U(x, t)])
V
N
<Q[U(x,
t)l>N,h---- ~ ,~,~. K[x--x(")(t),hlQ[U(")(t)].
(48)
For small h, the bias in this estimate is
((Q)s,h)
- (Q) = ~h2V2(Q) ¯
(49)
Hence, as h tends to zero, K(r, h) tends to 6(r) and ((Q)N.h) converges
(Q).
But as h becomessmaller, fewer particles have significant values of
K[x--x(")(t), h], and so the statistical error rises: The variance of
varies as
V/(Nh°) = (L/h)D/N,
(50)
whereL -= V l/D is a characteristic length of the domain.It is readily shown
that, for large N, it is optimal for h/L to vary as N- l/(4+D). For then, the
sumof the bias and the rms statistical error is minimized,each varying as
N-2/(4+~). Thus, for such a choice of h we have
lim
(Q)N,h
=
(Q).
(51)
These results have two major significances. First, Equation (51) shows
the convergenceof the particle representation. Second, it is likely that in
a numerical implementationthe error decreases with increasing N no faster
than N- 2/(4+0).
Althoughit is seldomdone in practice, it is in principle possible to use
the aboveideas to extract multi-time Lagrangianstatistics. It is important
to realize, however,that (at a fixed time) the particle representation contains no two-point information. Recall that different particles can be
viewed as being sampled from different, independent realizations of the
flow.
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5.
POPE
STOCHASTIC
LAGRANGIAN
MODELS
5.1 Generalized
Langevin Model
The Langevin equation for inhomogeneous flows described in Section 3
(Equation 32) is referred to as the Simplified Langevin Model(SLM).It
the simplest possible extension of the basic Langevin model(Equation 29)
that is consistent with momentum
and energy conservation.
The Generalized Langevin Model (GLM)--proposed by Pope (1983a)
and developed and demonstrated by Haworth & Pope (1986, 1987)overcomes some of the qualitative and quantitative defects of the SLM.
For the increment in the fluid-particle velocity U*(t), the GLM
dU~* = - 0447- ) dt+fgij(~*-(U~.))dt+(Co(e))t/2dWi,
Oxi
(52)
wherethe drift coefficient tensor (qij is a modeledfunction of the local mean
velocity gradients O(Ug)/Oxj,Reynoldsstresses (u~uj), and dissipation
It may be immediately observed that the SLMcorresponds to the simple
specification
(53
Hence the GLMis distinguished by a more elaborate specification of
The first term in Equation (52) is uniquely determined by the mean
momentumequations (at high Reynolds number, when the viscous term
is negligible). The final term in Equation (52) (the diffusion term) has
same form as in homogeneousisotropic turbulence. This is justified (at
high Reynolds number) by the Kolmogorov(1941) hypotheses: The term
pertains to small time-scale (high frequency) processes that are hypothesized to be locally isotropic and characterized by (e). (Implications
of the Kolmogorov1962 hypotheses are discussed in Section 5.3, and
Reynolds-numbereffects in 5.4.)
In the construction of the drift term (involving ~g~), the principal assumption madeis that the term is linear in U*. For homogeneous
turbulence, the
assumptionis fully justified, since this linearity is necessaryand sufficient
(Arnold 1974) for the joint pdf of velocity to be joint normal, in accord
with experimental observations (e.g. Tavoularis & Corrsin 1981).
The observed joint no~ality of the one-point velocity pdf in homogeneous turbulence leads to several important results. For this case, the
joint pdf is fully determined by the (known) mean velocity, and by the
covariance matrix, namely the Reynolds stresses; and, according to the
GLM(Equation 52), the Reynolds stresses evolve
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(54)
~ (u~u,)= ~,+fVk,(u,u,)fV,,(u~uk) +Co
(~),s~,,
where ~ is the production tensor:
~ ~ - (u~u~) Ox~ - (u~u~)o~).
(55)
Thus for any choice of ~q there is a corresponding modeled Reynoldsstress equation, which (as shownby Pope 1985) is realizable. The known
behavior of the Reynoldsstress equation in certain limits (e.g. rapid distortion, or two-component turbulence) can then be invoked to impose
constraints on
Using these constraints and experimental data on homogeneousturbulence, Haworth & Pope (1986) determined a specific form of if0 that
accurately describes the evolution of the Reynoldsstresses (and hence the
velocity joint pdf) in these flows. As an example, Figure 4 shows the
evolution of the anisotropy tensor
~ ~ (u~)/(u~u,)o,
(~6)
for the plane strain experiment of Gence & Mathieu (1979).
As is customary in turbulence modeling, with simplicity as the main
justification, the same model is used in inhomogeneousflows. Haworth&
0.2
0
0
A
A
0-
-
-0.2
0
0.01
0.02
Figure 4 Reynoldsstress anisotropies b~t =- (u, ut)/(ulul)--~6~ against time for transverse
plane strain of homogeneousturbulent. Symbols: experimental data of Genee & Mathieu
(1979) ~ b~ ~, ~ b:~, O b33, ¯ b2a. Lines: GLMcalculations. (From Haworth& Pope 1986.)
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POPE
Pope (1987) describe the successful application of the GLMto a range
free shear flows.
Finally, we observe that there is a Reynolds-stress equation corresponding to the SLM.Specifically, for homogeneousturbulence, Equations (52) and (53) lead
d
~ {uku,) - ~k,- (2 + 3(70) {e)dk,-- ~(e)~Sk,,
(57)
which is Rotta’s (1951) model. Thus, in Reynolds-stress-closure terminology, the GLM
is superior to the SLMin allowing for a nonlinear returnto-isotropy, and for incorporating "rapid pressure" effects. Recently
Pope (1993) considered in detail the relationship between the GLMand
Reynolds-stress models, and thereby deduced specifications corresponding to the isotropization of the production model (IPM, Naot et al 1970)
and to the SSGmodel(Speziale et al 1991).
Model for Frequency
5.2 Stochastic
The Generalized Langevin Model, just described, leads to a modeled
transport equation for the velocity joint pdff(V; x, t). This equation does
not provide a complete model, because the mean dissipation rate {e) (or
equivalent information) must be supplied separately--from a modeled
transport equation for (e), for example. This shortcoming motivated the
development of a complete closure based on the joint pdf of velocity and
dissipation (Pope & Chert 1990), which required the development of
stochastic modelfor dissipation.
In fact, rather than the instantaneous dissipation rate e(x, t) the model
developed by Pope &Chen(1990) is based on the turbulence frequency
defined by
co(x, t) = e(x, t)/k(x, t).
(58)
It should be noted that this is a mixedquantity in that e(x, t) is random
whereask(x, t) is not. Thus the probability distribution of co is the same
as that of e, to within a scaling. The mean frequency (~o) has been used
previously as a turbulence-model variable by, for example, Kolmogorov
(1942) and Wilcox (1988).
For homogeneousturbulence, Pope &Chen(1990) developed a stochastic modelco*(t) for the turbulent frequencyfollowinga fluid particle, co+(t).
Their model is constructed by rcference to the Lagrangian statistics of
dissipation extracted from direct numerical simulations by Yeung &Pope
(1989).
The simulations show that (to a very good approximation) the one-
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point one-time distribution of e is log-normal. That is, for fixed t, the
random variable
Z+(t) -= In [e+(t)/(e)] = In [co+(t)/(co)],
(59)
is Gaussian, with variance denoted by a 2. Further, except near the origin,
the autocorrelation function of Z+(t), p~(s), is well approximatedby the
exponential
px(s) -Islv*.
=e
(60)
where T~ is the corresponding integral time scale. The simulations support
the approximation
T~ -1
= Cx((D),
(61)
with Cz being a constant.
Giventhat ~ + (t) has a Gaussian pdf and an exponential autocorrelation,
it is obvious to model it as an OUprocess. The appropriate stochastic
differential equation is
dz*(t) = - [Z*(t) - (~*(t))] dt/T~ + (2~rZ/T~)’/2dW,
(cf Equation 21).
The modeled frequency is related to ~* by
~*°~,
co*(t) : (co(t))e
(62)
(63)
(cf Equation 59). Consequently, in order to obtain a model equation
for o~*, it is necessary also to model the evolution of (co). With the
nondimensional rate of change S~ defined by
dQo)= _ (co) 2S, ’
(64)
dt
the standard model equation for (e) (Launder & Spalding 1972) implies
S,o = (C,2-- 1)--(C,,- l)P/<e>,
(65)
where C,~ and C~2 are standard model constants, and P is the rate of
production of turbulence kinetic energy.
The stochastic model for co* proposed by Pope &Chen(1990) is then
obtained from Equations (62)-(64):
dco*=-co*(co) dt{ S~ + Cz [ln
(co*/<co))+co*(2C~(co)a2)~/2 (66)
The above development pertains to homogeneousturbulence. The extension of the model to inhomogeneousflows is considered by Pope (1991b).
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The only significant modification required to Equation (66) is the addition
of a term that (under appropriate circumstances) causes nonturbulent
fluid (characterized by co*= 0) to become turbulent ((o*> 0). As
scribed in the next subsection, with this modification, the stochastic
modelfor frequency is successful in describing the intermittent turbulent/
nonturbulent regions of free shear flows.
5.3 Refined Langevin Model
The stochastic model for frequency co*(t) (Equation 66) can be combined
with the Generalized Langevin Model (Equation 52) to provide a closed
modeled joint pdf equation. However,if the frequency co*(t) following
fluid particle is known,it is possible to incorporate this information in a
stochastic modelfor velocity so as to increase its physical realism. Sucha
refined Langevin model has been developed by Pope & Chen (1990) and
Pope (1991b).
According to all of the Langevin equations described above, for a small
time interval s (s/T << 1), the modeledLagrangian velocity increment
AsU*(t) U*(t+s)-U*(t),
(67)
is an isotropic Gaussian random vector with covariance
(AsUT(t)AsU~(t))=
Co<~e)st~ij-~-O(s2).
(68)
This covariance is consistent with the refined Kolmogorov(1962) hypotheses; but the Gaussianity of AsU*(t) is clearly at odds with notions
internal intermittency. In the spirit of Kolmogorov’srefined hypotheses,
it is natural to model AsU*(t) in terms of the particle dissipation
e*(t) = k~o*(t). This is simply achieved by replacing the diffusion coefficient
C0(e) in the Langevin equation by C0e*. Then, the conditional covariance
of AsU(t)
(AsUf(t)AsU~*(t)le*(t) = g) = Cogs6ij+O(s2),
while, correctly,
(69)
the unconditional variance is again given by Equation
(68).
Since the performance of the GLM
is completely satisfactory for homogeneous turbulence, Pope & Chen (1990) developed the Refined Lan#evin
model (RLM) to retain this behavior (while replacing Co(e) by
Coe* = Cokco* in the diffusion term). For homogeneousturbulence the
model is
dUff -where
~(P) dt+.L~,j(U~*-(U~))dt+(Cokog*)l/2dmi,
OX
i
(70)
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(71)
and the tensor ~ij is the inverse of (uiuj)/(~k). Notice that, comparedto
the GLM(Equation 52), the additional term in ~i~ is needed to produce
the correct Gaussian joint pdf of velocity (in homogeneousturbulence).
Additional modifications for inhomogeneousflows are described by Pope
(1991b).
The combination of the stochastic model for 09*(0 (Equation 66)
the RLMfor U*(t) provides a closed modeledevolution equation for their
joint pdf. This is the most advancedpdf modelcurrently available. It has
been applied to several different flows by Pope(1991b), Anandet al (1993),
and Norris (1993). Calculations for a plane mixing layer are nowbriefly
reported in order to illustrate several features of the model.
The calculations pertain to the statistically plane, two-dimensional,selfsimilar mixing layer formed between two uniform streams of different
velocities. The dominantflow direction is xl; the lateral direction is x2;
and the flow is statistically homogeneous
in the spanwise direction x3. The
free-stream velocities are U~o(at x2 = ~) and 2U~(at x2 = - ~), so that
the velocity ratio is 2, and the velocity difference is AU= Uo~. At large
axial distances the flow spreads linearly and is self-similar. Consequently,
statistics of U(x, t)/A U dependonly on x2/xl [where(x l, x2) = (0, 0) is
virtual origin of the mixing layer]. Lang (1985) provides experimental data
on this flow. The calculations are performed by integrating the stochastic
differential equationsfor the properties (x¢"), U¢"), o9¢"); n = 1,2 ..... N)
N ~ 50,000 particles. A comparison of the mean and rms velocities with
experimental data shows good agreement (see Pope 1991b).
Figure 5 is a scatter plot of the axial velocity and lateral position. For
this flow, with extremely high probability, there is no reverse flow (i.e.
U~l")(t) > 0 for all n and t). Hencethe axial location X~l")(t) of each particle
increases monotonically with time. Figure 5 is constructed by plotting the
points (U]")/AU, x~")/xl) for about one fifth of the particles (selected at
random) as they pass a particular axial location x~. (In view of selfsimilarity, the value of x~ is immaterial.)
At large and small values of x~/xl, the points are dense at U~*/AU= 1
and 2, respectively, and so appear as horizontal straight lines. Thesepoints
correspondto fluid with the free-stream velocity. At the center of the layer
(e.g. x~/xl = 0), the points are broadly scattered in U?/AU,indicative of
turbulent fluctuations with rms of order 0.2. Towardthe edges of the layer,
bimodal behavior is evident: with increasing distance from the layer, a
band of points tends to the free-stream velocity, while other points exhibit
fluctuations of order 0.1, but with decreasing probability. This reflects the
turbulent/nonturbulent nature of these regions.
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POPE
-0.05
0.0
0.05
Figure 5 Scatter plot of axial velocity and lateral position from joint pdf calculations of
the self-similar plane mixing layer (from Pope199 lb).
The intermittent nature of the edges of the mixing layer is yet more
evident in Figure 6, whichis a scatter plot of frequencyand lateral position.
The frequency is normalized by its maximummean value (~0)max (at
axial location considered) and is shown on a logarithmic scale. At the
edges of the layer the bimodalnature of~o*is clear: There is a diffuse band
of points centered around o~* ~ 0.3(co7 .... with a second denser band with
a)* values two or three orders of magnitudeless. These bands correspond to
turbulent and nonturbulent fluid respectively.
For inhomogeneousflows, experimental data on Lagrangian quantities
are essentially nonexistent. For this reason, there has been little impetus
to extract Lagrangian statistics from pdf calculations. However, as an
illustration of the type of information that is available in Lagrangian PDF
methods, shownon Figure 7 are the fluid particle paths of five particles
whoseinitial positions were selected at randomnear the center of the selfsimilar mixing layer. It maybe observed that several of these trajectories
traverse the layer monotonically, and that the trajectories are devofd of
high wave number fluctuations. From this we conclude that the motion
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I
10
I
i
-
. ~....:.. .. : ....: ":’... . : -..
~.:.. ::.. : ~.’, ...’.,::, ;’,:,~..:.; ; .;, ~"... ..
¯ ..;..?-.:;-.::.~.’:~:;.!,’,.~’~.
:..,:-’.:~.,.?.’d.~’:........
:..
.
~:: ¯ .-."~.~:~,.±.
~.~..7.9,
:.’.~,’.;¢.~.".;.
>:"- ...
~’~r"-V~~$~-z--,. ............
1 ............
~
101
~,V
~
49
_
". ....:.:-,:.~;,:~,.!.~.:.,,~-.,~,,,.,.-.,..:.....:,.:....
:Y,ar,E’q::t.’-,.~
".’;;;:.,~.~..:-.:">:
.-’."."’"
..-..."~]~:...-:::.:.-;::e:.~-.~.,.~-..-:....
...
:.~."
~’a::~ ’.’." ’.." "~..’-x~"- ".. "
_
.
¯ ~:.~.,.-,. ".
,. - ." ...~.,-~.~.,
¢;..~"¢’~:."-.’~-~.~:~."i
:-:"
-.:..
.... . ¯
104
I
0.0
-0.05
0.05
x~/x~
Figure 6 Scatter plot of turbulence frequency and lateral position from joint pdfcalculations
of the self-similar plane mixing layer (from Pope 1991b).
0. 10,
50.
zl
100.
Figure 7 Fluid particle paths in the self-similar plane mixing layer according to stochastic
models: x~ and x2 have arbitrary units. The dashed lines showthe nominal edgc of the layer,
wherethe meanvelocity differs from the free-stream velocity by 10%of th,e velocity difference.
(From Pope 1991b.)
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vow
implied by the model is consistent with the large-scale coherent motions
observed experimentally in mixing layers; and, conversely, it does not
resemble the small-scale randommotion analogous to molecular diffusion
or Brownian motion.
5.4 Stochastic
Model for Acceleration
Whenexamined in detail, the basic Langevin model (described in Section
3) can be justified on physical groundsonly in the limit of infinite Reynolds
number Re. Sawford (1991) presents a stochastic model for the fluid
particle acceleration which is extremely valuable and successful in incorporating Reynolds number effects. One virtue of the model is that it can
be directly related to Lagrangian statistics obtained from direct numerical
simulations--which are found to depend strongly on Reynolds number.
In the limit of infinite Reynoldsnumber,the modelreverts to the Langevin
equation. [As Sawfordshows, his model is equivalent to a different formulation given earlier by Krasnoff & Peskin (1971).]
As in Section 3 we consider stationary homogeneous isotropic turbulence with zero mean velocity. The turbulence is characterized by its
intensity u’ (or kinetic energy k = ~u’2), the meandissipation rate (e~,
by the kinematic viscosity v. In terms of these quantities, the Reynolds
numberis defined by:
2k
Re = (-~.
(72)
It is instructive to relate the Reynoldsnumberto time scales. As usual, the
eddy-turnover time TE and the Kolmogorovtim6 scale r,~ are defined by
Tz =- k/Q,) = ~2u’2/(e),
(73)
and
(74)
Hence we obtain
2.
Re = (Tz/v,)
(75)
The Langevin equation contains the single time scale T~; whereas
Sawford’s stochastic model for acceleration contains two time scales,
To and 3. These time scales (precisely defined below) scale as TE and
respectively, at high Reynolds number.
Let U*(t) and A*(t) denote the model for one component of velocity
and acceleration following a fluid particle. Thenthe velocity evolves by
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LAGRANGIAN PDF METHODS
d
~ U*(t) = A*(t).
(76)
With a’2 defined by
a"2 = u’2/(T~z),
(77)
Sawford’s stochastic modelfor acceleration can be written
(78)
where W(t) is a Wiener process.
Ananalysis of this model(see Sawford1991 or Priestly 1981) reveals that
U*(t) and A *(t) are stationary processes with zero meansand variances "2
and a’z, respectively. The autocorrelation function of U*(t) is
p*(s)=Ie-’"/r~--(~)e-’S’/~]/(1--~),
(79)
from which it follows that the (modeled) Lagrangian integral time scale
T= To~ + v.
(80)
The principal features of this model are most clearly seen at high (but
finite) Reynoldsnumber,at which there is a complete separation of scales,
i.e. ¯ << To~.For all times s muchlarger than z, the velocity autocorrelation
function is p*(s) exp(--[sl/T~)--the sa me asfor the Langevin mode
Consequently,in the inertial range (z << s << To), the Lagrangianvelocity
structure function varies linearly with s, in accord with the Kolmogorov
hypotheses (Equations 25-27). Correspondingly, the Lagrangian velocity
frequency spectrum varies as ~o-2. But for times s comparable to z, this
model is quite different from the Langevin model. Because U*(t) is a
differentiable function of time, the autocorrelation function has zero slope
at the origin. For not-too-large s/z, the autocorrelation function of acceleration is p~,(s) ~-, exp (-Isl/~). Correspondingly,the Lagrangianvelocity
frequency spectrum varies as ~o-4 at high frequency (~or >> 1).
In order to complete the model, two specifications are required to fix
T~o and z in terms of TE and Re. Sawford (1991) used the Lagrangian DNS
data of Yeung& Pope (1989) to achieve this. Here we do the same, but
a slightly different way. First, the DNSdata on the Kolmogorov-scaled
acceleration variance
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r, ov~
ao = a’2"~n/(e),
(81)
can be well approximated (for not too small Ra)
ao ~ 3(1-22/Ra),
(82)
where R~ = (20/3Re)’/2 is the Taylor-scale Reynolds number. [This form
of correlation can be justified in terms of the inertial-range pressure fluctuation spectrum (M. S. Nelkin 1991, private communication; George et
al 1984).] Second, for each value of Rz studied in the DNS,the quantity
4TE
Cr(R~) = 3 T~’
(83)
can be determined by m~ttching T/r, between DNSand the model. The
values obtained are between 6 and 7, with a least-squares fit yielding
Cr ~ Cx(o~) (1 +4/Ra),
(84)
with Cx(oo) = 6.2. In this case there is no justification for the form of the
correlation, and the data exhibit significant scatter around it. Given the
empirical correlations for ao and CTthe two time scales are determined as
T~ = T~
,
(85)
and
r= "\2aoJ"
(86)
The ability of this modelto describe Lagrangianstatistics is impressive.
Figure 8 showsa comparison of the acceleration autocorrelation functions
pA(S) obtained from the model and from DNS. The agreement indicates
that the model provides a good approximation to the short-time behavior
[although, because A*(t) is not differentiable, p~(s) has finite slope at the
origin].
A revealing plot is of the Lagrangian velocity structure function DL(S)
(Equation 25) normalized by. (e)s. As may be seen from Figure 9,
model is in good agreement with the DNSdata, and correctly shows that
the peak value--denoted by C0*--increases with R~. According to the
Kolmogorovhypotheses, at high Reynolds number, and for inertial-range
times s (z, << s << T), the quantity D~(s)/((~)s), adopts a constant value
C0. It is readily shownthat the modelhas this property, with C0 = Cx(~).
But, as may be seen on Figure 10, the peak value Cg of DL(S)/((e)s)
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LAGRANGIAN PDF METHODS
1.0
I
I
53
I
0.8
0.6
’~ 0.4
0.0I- - %*_,
- - -. - .7,,~7~.---~
0
5
10
15
20
Figure 8 Acceleration autocorrelation function against Kolmogorov-scaled time lag. Symbols: DNSdata (Yeung & Pope 1989); lines: Sawford’s model. Ra = 38: ~ ---; R~ ~ 63:
~ ~; R~ = 90: ~ ~; R~ = 93: ~. (From Sawford 1991, with permission.)
4
~
~/
o ,~’,o,
9~’.i
10 -2
10 1
1
~~
10
102
103
Figure 9 Lagrangian velocity structure function DL(S) divided by (e)s against Kolmogorovscaled time, s/%. Symbolsand lines, same as Figure 8. (FromSawford1991, with permission.)
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54
POPE
7.0
i
Co
4.0
3.0
2.0
2
10
~
10
4
11)
5
10
6
10
Figure 10 Co*[the peakvalueof DL(s)/(e)s]againstTaylor-scaleReynolds
number.
Symbols: DNSdata (Yeung & Pope 1989); full line: from stochastic model for acceleration,
dashed line: model asymptote
approaches Co slowly as R~ increases: At the relatively high value
Rz = 1000, Co* is only 85%of Co.
Figure 11 showsthe ratio of the Lagrangian to Eulerian time scales. It
maybe seen that this ratio varies appreciably over the range of R~accessible
to DNSand wind tunnel experiments.
A question of some interest and importance is the value of the
Kolmogorov constant
Co. The estimate
from the above model
[Co = CT(~) = 6.2] is, in essence, obtained by extrapolating from DNS
data in the R~ range 40-90. Other values given in the literature are:
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LAGRANGIAN PDF METHODS
55
0.4
T/T
E
30
100
300
1000
Rx
Figure 11 Ratio of Lagrangian to Eulerian time scales against Taylor-scale Reynolds
number. Symbols: DNS(Yeung & Pope 1989); line: stochastic model for acceleration,
Equations (83 86).
Co ~ 3.8_+ 1.9 from measurements in the atmospheric boundary layer
(Hanna 1981); Co ~ 5.0 from kinematic simulations (Fung et al 1992);
Co ~ 5.9 from the Lagrangian renormalized approximation theory
(Kaneda 1992); and Co = 5.7 based on the Langevin equation and further
assumptions applied to the constant-stress region of the neutral atmospheric boundary layer (Rodean 1991). It is not unreasonable to suppose,
therefore, that Co is in the range 5.0-6.5.
With the Langevin and refined Langevin models (which contain no
Reynolds-numberdependence) it is found that values of Co = 2.1 (Anand
&Pope 1985) and Co = 3.5 (Pope &Chen1990), respectively, are required
to calculate accurately the dispersion behind a line source in grid turbulence
at Ra ~ 70. It is nowapparent that these values--while being appropriate
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56
POPE
values of the model constants at moderate Reynolds number--do not
correspond to the value of the Kolmogorovconstant Co.
A stochastic model for the fluid p~.rticle acceleration A*(t) (such
Equation 78), combined with the equations ~* = U* and 1]* = A*, leads
to a modeled equation for the Eulerian joint pdf of velocity and acceleration, J~A. Such a model equation has not, to date, been applied to
inhomogeneousflows. Comparedto the velocity joint pdf equation (stemming from a Langevin equation), the equation forf~A has the advantages of
incorporating Reynolds-number effects and of representing Kolmogorovscale processes. This maybe of particular value in the study of near-wall
flows.
5.5
Other Stochastic
Models
Table 1 summarizes the stochastic Lagrangian models that have been
proposed for various fluid properties.
Table 1 Stochastic
Lagrangian models of turbulence
Subject of model
Authors
Fluid particle position
Taylor (1921
Fluid particle velocity
(single-particle dispersion)
Novikov (1963), Chung (1969), Frost (1975),
(1979), Wilsonet al (1981 a~z), Legg &Raupach(1982),
Durbin (1933), Ley & Thomson(1983), Wilson
al (1983), Thomson(1984), Anand & Pope (1985),
Dopet al (ii985), De Baas et al (1986), Haworth
Pope ( 198611, Sawford (1986), Thomson(1986a),
(1987), Thomson(1987), Maclnnes & Bracco (1992)
Fluid particle acceleration
Sawford( 199 I)
Relative velocity between
fluid particle pairs
(two-particle dispersion)
Novikov (1963), Durbin (1980), Lamb (1981), Gifford
(1982), Sawford (I 982), Durbin (1982), Lee &
(1983), Sawford & Hunt (1986), Thomson(1986b),
Thomson (1990)
Dissipation
Pope &Chen(1990), Pope (I 991
Velocity-gradient tensor
Pope & Cheng (1988), Girimaji & Pope (1990)
Scalar (e.g. species
concentration)
Valifio & Dopazo (1991)
Scalar and scalar gradient
Fox (1992)
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Taylor (1921) proposed a stochastic model for a componentof fluid particle position x*(t) in whichsuccessive increments Ax*~ x*(t + At)- x*(t)
are correlated. It is interesting to observe that the statistics implied by
this model are identical to those from the Langevin equation (Durbin
1980).
For the fluid particle velocity U*(t), early proposals for the use of the
Langevin equation were made by Novikov (1963), Chung (1969),
Frost (1975). The works cited in Table 1 from the period 1979-1984reflect
active use of stochastic modelingof atmospheric dispersion. These models
are essentially of the Langevin type, with the primary issue being the
specification of the coefficients. In inhomogeneous
flows, if the coefficients
are specified incorrectly, stochastic modelscan predict (incorrectly) that
an initially uniform distribution of particles becomesnonuniform. Most
of the works since 1985 address this issue; a complete explanation is
provided by Pope (1987).
The concentration variance of a contaminant in a turbulent flow can be
studied in terms of the relative dispersion of fluid particle pairs (Batchelor
1952). Hencestochastic models have been developed (see Table 1) for
relative velocity between particles. These models have had somenotable
successes in predicting and explaining experimental observations. For
example, Durbin (1982) shows that a two-particle dispersion model
accounts for the observed sensitivity of the scalar variance in decaying
grid turbulence to the initial scalar-to-velocity length scale ratio; and
Thomson(1990) shows that his modelaccounts for the nontrivial evolution
of the correlation coefficient betweenscalars emanatingfrom a pair of line
sources in grid turbulence, which has been studied experimentally by
Warhaft (1984).
A key quantity in the specification of two-particle modelcoefficients is
the separation distance between the particles. Only recently (Yeung1993)
have DNSresults that can be used to develop and test such models become
available. Someinsight is also provided by kinematic simulations (Fung
et al 1992).
The local deformation of material lines, surfaces, and volumes in a
turbulent flow is determined by the velocity gradient tensor following the
fluid (see e.g. Monin &Yaglom1985). This motivated the development
stochastic Lagrangian models for the velocity gradient tensor by Pope &
Cheng (1988) and Girimaji & Pope (1990). One use of such models is
the calculation of the area density of premixedturbulent flame sheets (Pope
& Cheng 1988).
An important yet difficult topic is stochastic Lagrangian models~b*(t)
for a set of scalars ~b ÷ (t)--such as temperatureand species concentrations.
In conjunction with a Langevinmodel, a stochastic modelfor ~b*(t) leads
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58
POPE
to a modeled equation :for the Eulerian joint pdf of velocity and composition which can be used to study turbulent reactive flows. Examplesof
applications of this approach can be found in Anand&Pope (1987), Masri
& Pope (1990), Haworth & E1 Tahry (1991), Correa & Pope (1992),
(1993), Taing et al (1993), and elsewhere.
A set of compositions q~ has certain properties that are very different
from these of velocity U. Amongthese are: boundedness; localness of
interactions in composition space; and (in important limiting cases) linearity and independence (Pope 1983b, 1985). These properties make the
modelingof 4~*(t) different and more difficult than the modelingof U*(t).
Currently there is no modelthat is even qualitatively satisfactory in all
respects.
The simplest model---proposed in several different contexts and with
different justifications--is the linear deterministic model:
dt
- C4(co) (~b* - (q~))
(87)
(Chung 1969, Yamazaki & Ichigawa 1970, Dopazo & O’Brien 1974, Frost
1975, Borghi 1988). Although (in application to inhomogeneousturbulent
reactive flows) the modelis not without merit, because it is deterministic,
it clearly provides a poor representation of time series of the fluid particle
composition ~b+(t).
Also widely employed are stochastic mixing models (e.g. Curl 1963,
Dopazo 1979, Janicka et al 1979, Pope 1982). In the terminology of
stochastic processes, these modelsare point processes: The value of ~p*(t)
is piecewise constant, changing discontinuously at discrete time points.
Again, these modelshave their uses, but clearly the time series they generate, 4’*(0, are qualitatively different to those of turbulent fluid, 4,+(t).
Shownin Table 1 are the only proposed models that are stochastic, that
generate continuous time series, and that preserve the boundedness of
scalars. It is possible that a completely satisfactory model at this level
will not be achieved. Instead, it may be necessary to incorporate more
information, particularly that pertaining to scalar gradients (see e.g.
Meyers & O’Brien 1981, Pope 1990, Fox 1992).
6.
CONCLUSION-
Lagrangian PDF methods are based on stochastic Lagrangian models-that is, stochastic models for the evolution of properties following fluid
particles. For example, stochastic models for the fluid particle velocity
U*(t) (Equation 70) and for the turbulence frequency o~*(t) (Equation 66)
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59
lead to closed modelequations for both the Lagrangian and Eulerian joint
pdfs of these quantities. The Eulerian pdf equation can be used as a
turbulence modelto calculate the properties of inhomogeneousturbulence
flows. This equation is solved numerically
by a Monte Carlo method which
is based, naturally,
on the tracking of a large number of particles.
The primary stochastic
models reviewed here are for velocity
(based on
the Langevinequation), for the turbulent frequency (or dissipation),
for the fluid particle acceleration. Lagrangian statistics extracted from
direct numerical simulations of homogeneousturbulence have played a
central role in the developmentof these models. Similar statistics at higher
Reynolds numbers and in inhomogeneousflows are needed to develop and
test the modelsfurther.
Other fluid properties--most importantly the composition S--can be
adjoined to the PDFmethod. This requires stochastic models for the
quantities involved. In spite of considerable efforts, deficiencies remainin
stochastic models for composition.
There is a close connection between Lagrangian PDF methods and
Reynolds-stress closures. This connection can be used to benefit both
approaches. In particular, new ideas in Reynolds-stress modeling (e.g.
Durbin 1991, Lumley 1992, Reynolds 1992) can be readily incorporated
in PDF methods.
ACKNOWLEDGMENTS
I amgrateful to Dr. B. L. Sawfordfor permission to reproduce Figures 8
and 9.
For commentsand suggestions on the draft of this paper I thank M. S.
Anand, R. O. Fox, D. C. Haworth, J. C. R. Hunt, B. L. Sawford, D. J.
Thomson, C. C. Volte, and P. K. Yeung.
This workwas supported in part by the USAir Force Office of Scientific
Research (grant number AFOSR-91-0184),and by the National Science
Foundation (grant number CTS-9113236).
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